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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Sensitivity to initial conditions</h3>

<p class="header_title">Introduction</p>

<p>The idea of this simulation is to show that the trajectory of a system of particles is very sensitive 
to its initial conditions.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;In general, an isolated system of many particles that is prepared in a nonrandom configuration 
will change in time so as to approach its most random configuration where it is in equilibrium. What happens if we choose 
the initial conditions in a very special way?</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The default initial condition corresponds to N = 11 particles that are equally spaced vertically and all traveling with 
the same velocity to the right. The program solves Newton's second law of motion  numerically for each of the particles assuming 
the Lennard-Jones potential.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;In the second part of the simulation, a small perturbation is applied to particle 6, the one in the middle of the display. Its velocity 
in the horizontal direction is changed from 1.0 to 0.99 and its velocity in the vertical direction is changed from 0 to 
0.01. In the questions we will explore what happens and why.</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.sensitive.LJgasApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Problems</p>

<ol>

<li>Run the program with the default initial condition and describe the motion of the particles.</li>

<li>After the simulation proceeds for a while, click on the <tt>Perturb</tt> button. What happens to the motion of the particles? After the simulations proceeds for a while, say t = 0.02, stop the simulation and click the <tt>Reverse</tt> button, which 
reverses the velocities of all particles. Does the system return to its initial special state? Is the motion reversible?</li>

<li>Proceed as in Problem 2, but wait longer before 
perturbing the system. Does the system return to its initial state. Why or why not?</li>

</ol>

<p class="header_title">Java Classes</p>
<ul>
<li>LJgas</li>
<li>LJgasApp</li>
</ul>

<p class = "small">Updated 28 February 2007.</p>
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